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A q-microscope for supercongruences

 报告题目： A q-microscope for supercongruences 报 告 人： 郭军伟 教授(淮阴师范学院) 报告时间： 2018年05月28日 15:00--16:00 报告地点： 理学院东北楼四楼报告厅（404） 报告摘要： By examining asymptotic behavior of certain infinite basic (\$q\$-) hypergeometric sums at roots of unity (that is, at a `\$q\$-microscopic' level) we prove polynomial congruences for their truncations. The latter reduce to non-trivial (super)congruences for truncated ordinary hypergeometric sums, which have been observed numerically and proven rarely. A typical example includes derivation, from a \$q\$-analogue of Ramanujan's formula \$\$sum_{n=0}^inftyfrac{binom{4n}{2n}{binom{2n}{n}}^2}{2^{8n}3^{2n}},(8n+1)=frac{2sqrt{3}}{pi},\$\$ of the two supercongruences \$\$S(p-1)equiv pbiggl(frac{-3}pbiggr)pmod{p^3} quadtext{and}quad SBigl(frac{p-1}2Bigr) equiv pbiggl(frac{-3}pbiggr)pmod{p^3},\$\$ valid for all primes \$p>3\$, where \$S(N)\$ denotes the truncation of the infinite sum at the \$N\$-th place and \$bigl(frac{-3}{cdot}bigr)\$ stands for the quadratic character modulo~\$3\$.

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